Información Básica Para El Diseño De Las PCH

A polygon is said to be regular if all its sides are of equal length and all its internal angles are equal. We call a regular polygon with n sides a regular n-

gon. Some of these shapes are probably quite familiar; for example, a regular n-gon with n = 3 is just an equilateral triangle, n = 4 is a square, n = 5 is a

regular pentagon, and so on:

A polyhedron is regular if its faces are regular polygons, all with the same number of sides, and also each vertex belongs to the same number of edges.

Three examples of regular polyhedra come more or less readily to mind: the cube, the tetrahedron and the octahedron. These are three of the famous five

Platonic solids; the other two are the less obvious icosahedron, which has 20

triangular faces, and dodecahedron, which has 12 pentagonal faces. Here are pictures of the octahedron, icosahedron and dodecahedron:

Every regular polyhedron carries five associated numbers: three are V, E,

78 A CONCISE INTRODUCTION TO PURE MATHEMATICS

edges each vertex belongs to. We record these numbers for the Platonic solids:

V E F n r tetrahedron 4 6 4 3 3 cube 8 12 6 4 3 octahedron 6 12 8 3 4 icosahedron 12 30 20 3 5 dodecahedron 20 30 12 5 3

As you might have guessed from the name, the Platonic solids were known to the Greeks. They are the most symmetrical, elegant and robust of solids, so it is natural to look for more regular polyhedra. Remarkably, though perhaps disappointingly, there are no others. This fact is another theorem of the great Euler. The proof is a wonderful application of Euler’s formula 9.1. Here it is.

THEOREM 9.3

The only regular convex polyhedra are the five Platonic solids.

PROOF Suppose we have a regular polyhedron with parametersV,

E, F, n and r as defined above.

First we need to show some relationships between these parameters. We shall prove first that

2E = nF . (9.1)

To prove this, let us calculate the number of pairs

e, f

where e is an edge, f is a face, and e lies on f . Well, there are E possibilities for the edgee, and each lies in 2 faces f ; so the number of such pairse, f is equal to 2E. On the other hand, there are F possibilities for the face f , and each has n edges e; so the number of such pairs e, f is also equal tonF. Therefore, 2E = nF, proving (9.1).

Next we show that

2E = rV . (9.2)

The proof of this is quite similar: count the pairs

v,e

wherev is a vertex, e an edge, and v lies on e. There are E edges e, and each has 2 verticesv, so the number of such pairs v,e is 2E; on the other hand, there areV vertices v, and each lies on r edges, so the number of such pairs is alsorV. This proves (9.2).

EULER’S FORMULA AND PLATONIC SOLIDS 79 At this point we appeal to Euler’s formula 9.1:

V −E +F = 2 .

SubstitutingV =2E_r ,F =2E_n from (9.1) and (9.2), we obtain 2E_r −E +2E_n = 2; hence

1

r+1n=12+E1 . (9.3)

Now we know that n ≥ 3, as a polygon must have at least 3 sides; likewise r ≥ 3, since it is geometrically clear that in a polyhedron a vertex must belong to at least 3 edges. By (9.3), it certainly cannot be the case that bothn ≥ 4 and r ≥ 4, since this would make the left-hand side of (9.3) at most 1₂, whereas the right-hand side is more than 1₂. It follows that eithern = 3 or r = 3.

Ifn = 3, then (9.3) becomes 1

r =16+E1 .

The right-hand side is greater than 1₆, and hencer < 6. Therefore, r = 3, 4 or 5 andE = 6, 12 or 30, respectively. The possible values of V, F are given by (9.1) and (9.2).

Likewise, if r = 3, (9.3) becomes 1_n =1₆+_E1, and we argue similarly thatn = 3,4 or 5 and E = 6,12 or 30, respectively.

We have now shown that the numbersV,E,F,n,r for a regular poly- hedron must be one of the possibilities in the following table:

V E F n r 4 6 4 3 3 8 12 6 4 3 6 12 8 3 4 12 30 20 3 5 20 30 12 5 3

These are the parameter sets of the tetrahedron, cube, octahedron, icosa- hedron and dodecahedron, respectively. To complete the proof we now only have to show that each Platonic solid is the only regular solid with its particular parameter set. This is a simple geometric argument, and we present it just for the last parameter set — the proofs for the other sets are entirely similar.

So suppose we have a regular polyhedron with 20 pentagonal faces, each vertex lying on 3 edges. Focus on a particular vertex. At this vertex there is only one way of fitting three pentagonal faces together:

80 A CONCISE INTRODUCTION TO PURE MATHEMATICS

At each of the other vertices of these three pentagons, there is likewise only one way of fitting two further pentagons together. Carrying on this argument with all new vertices, we see that there is at most one way to make a regular solid with these parameters. Since the dodecahedron is such a solid, it is the only one. This completes the proof.

Exercises for Chapter 9

1. Consider a convex polyhedron, all of whose faces are squares or regular pentagons. Say there are m squares and n pentagons. Assume that each vertex lies on exactly 3 edges.

(a) Show that for this polyhedron, the following equations hold: 3V = 2E, 4m+5n = 2E, m+n = F . (b) Using Euler’s formula, deduce that 2m+n = 12.

(c) Find examples of such polyhedra for as many different values of m as you can.

2. Prove that for a convex polyhedron with V vertices, E edges and F faces, the following inequalities are true:

2E ≥ 3F and 2E ≥ 3V . Deduce using Euler’s formula that

2V ≥ F +4, 3V ≥ E +6, 2F ≥ V +4 and 3F ≥ E +6 . Give an example of a convex polyhedron for which all these inequalities are equalities (i.e., 2V = F +4, etc.).

3. Prove that if a connected plane graph has v vertices and e edges, and

v ≥ 3, then e ≤ 3v−6.

4. Prove that it is impossible to make a football out of exactly 9 squares and m octagons, where m ≥ 4. (In this context, a “football” is a convex polyhedron in which at least 3 edges meet at each vertex.)

5. Prove that if a finite connected plane graph has no faces, then it has a vertex that is joined to exactly one other vertex. (Hint: Assume for a contradiction that every vertex is joined to at least two others. Try to use this to show there must be a face.)

EULER’S FORMULA AND PLATONIC SOLIDS 81 6. Draw all the connected plane graphs with 4 edges, and all the connected

plane graphs with 4 vertices.

7. Let Kndenote the graph with n vertices in which any two vertices are joined by an edge. So, for example, K2consists of 2 vertices joined by

an edge and K3is a triangle.

Prove that it is possible to draw K4as a plane graph.

Prove that it is impossible to draw K5as a plane graph. (Hint: Use the

inequality in Exercise 3 cleverly.)

8. Prove that every connected plane graph has a vertex that is joined to at most five other vertices. (Hint: Assume every vertex is joined to at least 6 others, and try to use Exercise 3 to get a contradiction.)

9. Critic Ivor Smallbrain has been thrown into prison for libelling the great film director Michael Loser. During one of his needlework classes in prison, Ivor is given a pile of pieces of leather in the shapes of regular pentagons and regular hexagons and is told to sew some of these together into a convex polyhedron (which will then be used as a football). He is told that each vertex must lie on exactly 3 edges. Ivor immediately exclaims, “Then I need exactly 12 pentagonal pieces!”

Chapter 10